├── .gitignore
├── overview.png
├── numberings.png
├── demo
    ├── CMakeLists.txt
    └── main.cpp
├── lib
    ├── Triangulation.h
    ├── IO.h
    ├── CMakeLists.txt
    ├── Cube.h
    ├── DataStructs.h
    ├── IO.cpp
    ├── MeshReconstruction.h
    ├── MeshReconstruction.cpp
    ├── Cube.cpp
    └── Triangulation.cpp
├── CMakeLists.txt
└── Readme.md


/.gitignore:
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1 | bin
2 | html/
3 | latex/
4 | 


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/overview.png:
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https://raw.githubusercontent.com/Magnus2/MeshReconstruction/HEAD/overview.png


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/numberings.png:
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https://raw.githubusercontent.com/Magnus2/MeshReconstruction/HEAD/numberings.png


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/demo/CMakeLists.txt:
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1 | include_directories("." ${PROJECT_SOURCE_DIR}/lib)
2 | add_executable(Demo main.cpp)
3 | target_link_libraries (Demo MeshReconstruction)


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/lib/Triangulation.h:
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 1 | #pragma once
 2 | #include "Cube.h"
 3 | #include "DataStructs.h"
 4 | 
 5 | namespace MeshReconstruction
 6 | {
 7 | 	void Triangulate(
 8 | 		IntersectInfo const& intersect,
 9 | 		Fun3v const& grad,
10 | 		Mesh& mesh);
11 | }


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/lib/IO.h:
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1 | #pragma once
2 | #include <string>
3 | #include "DataStructs.h"
4 | 
5 | namespace MeshReconstruction
6 | {
7 | 	/// Writes a mesh to a file in <a href="https://de.wikipedia.org/wiki/Wavefront_OBJ">obj format</a>.
8 | 	void WriteObjFile(Mesh const& mesh, std::string const& file);
9 | }


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/lib/CMakeLists.txt:
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 1 | add_library(MeshReconstruction MeshReconstruction.h MeshReconstruction.cpp Cube.h Cube.cpp DataStructs.h IO.h IO.cpp Triangulation.h Triangulation.cpp)
 2 | 
 3 | install(TARGETS MeshReconstruction
 4 |         RUNTIME DESTINATION bin
 5 |         LIBRARY DESTINATION lib
 6 |         ARCHIVE DESTINATION lib)
 7 | 
 8 | install(DIRECTORY .
 9 |         DESTINATION .
10 |         FILES_MATCHING PATTERN "*.h")


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/CMakeLists.txt:
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 1 | cmake_minimum_required (VERSION 3.0)
 2 | project (MeshReconstruction)
 3 | 
 4 | set(CMAKE_CXX_STANDARD 11)
 5 | set(CMAKE_CXX_STANDARD_REQUIRED ON)
 6 | 
 7 | # Parallel compilation.
 8 | if(WIN32)
 9 | 	set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} /MP")
10 | 	set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} /wd4250")
11 | endif()
12 | 
13 | set(CMAKE_DEBUG_POSTFIX "d")
14 | 
15 | add_subdirectory(lib)
16 | add_subdirectory(demo)


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/demo/main.cpp:
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 1 | #include <MeshReconstruction.h>
 2 | #include <IO.h>
 3 | 
 4 | using namespace MeshReconstruction;
 5 | 
 6 | int main()
 7 | {
 8 | 	auto sphereSdf = [](Vec3 const& pos)
 9 | 	{
10 | 		auto const Radius = 1.0;
11 | 		return pos.Norm() - Radius;
12 | 	};
13 | 
14 | 	Rect3 domain;
15 | 	domain.min = { -1.1, -1.1, -1.1 };
16 | 	domain.size = {2.2, 2.2, 2.2};
17 | 
18 | 	Vec3 cubeSize{ 0.05, 0.05, 0.05 };
19 | 
20 | 	auto mesh = MarchCube(sphereSdf, domain, cubeSize);
21 | 	WriteObjFile(mesh, "sphere.obj");
22 | }


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/lib/Cube.h:
--------------------------------------------------------------------------------
 1 | #pragma once
 2 | #include "DataStructs.h"
 3 | 
 4 | namespace MeshReconstruction
 5 | {
 6 | 	struct IntersectInfo
 7 | 	{
 8 | 		// 0 - 255
 9 | 		int signConfig;
10 | 
11 | 		// If it exists, vertex on edge i is stored at position i.
12 | 		// For edge numbering and location see numberings.png.
13 | 		std::array<Vec3, 12> edgeVertIndices;
14 | 	};
15 | 
16 | 	class Cube
17 | 	{
18 | 		Vec3 pos[8];
19 | 		double sdf[8];
20 | 
21 | 		Vec3 LerpVertex(double isoLevel, int i1, int i2) const;
22 | 		int SignConfig(double isoLevel) const;
23 | 
24 | 	public:
25 | 		Cube(Rect3 const& space, Fun3s const& sdf);
26 | 
27 | 		// Find the vertices where the surface intersects the cube.
28 | 		IntersectInfo Intersect(double isoLevel = 0) const;
29 | 	};
30 | }


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/Readme.md:
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 1 | # MeshReconstruction
 2 | 
 3 | This is a small library that can reconstruct a triangle mesh from a <a href="http://www.iquilezles.org/www/articles/distfunctions/distfunctions.htm">signed distance function</a> using the <a href="https://en.wikipedia.org/wiki/Marching_cubes">Marching Cubes algorithm</a> and export it to a file in the <a href="https://de.wikipedia.org/wiki/Wavefront_OBJ">obj format</a>.
 4 | 
 5 | The library is self-contained and has no dependencies. The library is fast due to precomputed lookup tables and a narrow-band approach which excludes a lot of marching cubes that are far away from the surface.
 6 | 
 7 | The library requires C++14 and has been tested under Visual Studio 2017 and Windows 10 but should port to other systems without major problems.
 8 | 
 9 | The library can be used under the terms of the MIT License.
10 | 
11 | <p align="center">
12 |   <img src="overview.png" width="400" alt="Overview">
13 | </p>


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/lib/DataStructs.h:
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 1 | #pragma once
 2 | #include <vector>
 3 | #include <array>
 4 | #include <functional>
 5 | 
 6 | namespace MeshReconstruction
 7 | {
 8 | 	struct Vec3
 9 | 	{
10 | 		double x, y, z;
11 | 
12 | 		Vec3 operator+(Vec3 const& o) const
13 | 		{
14 | 			return { x + o.x, y + o.y, z + o.z };
15 | 		}
16 | 
17 | 		Vec3 operator-(Vec3 const& o) const
18 | 		{
19 | 			return { x - o.x, y - o.y, z - o.z };
20 | 		}
21 | 
22 | 		Vec3 operator*(double c) const
23 | 		{
24 | 			return { c*x, c*y, c*z };
25 | 		}
26 | 
27 | 		double Norm() const
28 | 		{
29 | 			return sqrt(x*x + y*y + z*z);
30 | 		}
31 | 
32 | 		Vec3 Normalized() const
33 | 		{
34 | 			auto n = Norm();
35 | 			return { x / n, y / n, z / n };
36 | 		}
37 | 	};
38 | 
39 | 	struct Rect3
40 | 	{
41 | 		Vec3 min;
42 | 		Vec3 size;
43 | 	};
44 | 
45 | 	using Triangle = std::array<int, 3>;
46 | 
47 | 	struct Mesh
48 | 	{
49 | 		std::vector<Vec3> vertices;
50 | 		std::vector<Triangle> triangles;
51 | 		std::vector<Vec3> vertexNormals;
52 | 	};
53 | 
54 | 	using Fun3s = std::function<double(Vec3 const&)>;
55 | 	using Fun3v = std::function<Vec3(Vec3 const&)>;
56 | }


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/lib/IO.cpp:
--------------------------------------------------------------------------------
 1 | #include "IO.h"
 2 | using namespace std;
 3 | 
 4 | void MeshReconstruction::WriteObjFile(Mesh const& mesh, string const& fileName)
 5 | {
 6 | 	// FILE faster than streams.
 7 | 
 8 | 	FILE* file;
 9 | 	auto status = fopen_s(&file, fileName.c_str(), "w");
10 | 	if (status != 0)
11 | 	{
12 | 		throw runtime_error("Could not write obj file.");
13 | 	}
14 | 
15 | 	// write stats
16 | 	fprintf(file, "# %d vertices, %d triangles\n\n",
17 | 		static_cast<int>(mesh.vertices.size()),
18 | 		static_cast<int>(mesh.triangles.size()));
19 | 
20 | 	// vertices
21 | 	for (auto vi = 0; vi < mesh.vertices.size(); ++vi)
22 | 	{
23 | 		auto const& v = mesh.vertices.at(vi);
24 | 		fprintf(file, "v %f %f %f\n", v.x, v.y, v.z);
25 | 	}
26 | 
27 | 	// vertex normals
28 | 	fprintf(file, "\n");
29 | 	for (auto ni = 0; ni < mesh.vertices.size(); ++ni)
30 | 	{
31 | 		auto const& vn = mesh.vertexNormals.at(ni);
32 | 		fprintf(file, "vn %f %f %f\n", vn.x, vn.y, vn.z);
33 | 	}
34 | 
35 | 	// triangles (1-based)
36 | 	fprintf(file, "\n");
37 | 	for (auto ti = 0; ti < mesh.triangles.size(); ++ti)
38 | 	{
39 | 		auto const& t = mesh.triangles.at(ti);
40 | 		fprintf(file, "f %d//%d %d//%d %d//%d\n",
41 | 			t[0] + 1, t[0] + 1,
42 | 			t[1] + 1, t[1] + 1,
43 | 			t[2] + 1, t[2] + 1);
44 | 	}
45 | 
46 | 	fclose(file);
47 | }


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/lib/MeshReconstruction.h:
--------------------------------------------------------------------------------
 1 | #pragma once
 2 | #include "DataStructs.h"
 3 | 
 4 | namespace MeshReconstruction
 5 | {
 6 | 	/// Reconstructs a triangle mesh from a given signed distance function using <a href="https://en.wikipedia.org/wiki/Marching_cubes">Marching Cubes</a>.
 7 | 	/// @param sdf The <a href="http://www.iquilezles.org/www/articles/distfunctions/distfunctions.htm">Signed Distance Function</a>.
 8 | 	/// @param domain Domain of reconstruction.
 9 | 	/// @returns The reconstructed mesh.
10 | 	Mesh MarchCube(
11 | 		Fun3s const& sdf,
12 | 		Rect3 const& domain);
13 | 
14 | 	/// Reconstructs a triangle mesh from a given signed distance function using <a href="https://en.wikipedia.org/wiki/Marching_cubes">Marching Cubes</a>.
15 | 	/// @param sdf The <a href="http://www.iquilezles.org/www/articles/distfunctions/distfunctions.htm">Signed Distance Function</a>.
16 | 	/// @param domain Domain of reconstruction.
17 | 	/// @param cubeSize Size of marching cubes. Smaller cubes yields meshes of higher resolution.
18 | 	/// @param isoLevel Level set of the SDF for which triangulation should be done. Changing this value moves the reconstructed surface.
19 | 	/// @param sdfGrad Gradient of the SDF which yields the vertex normals of the reconstructed mesh. If none is provided a numerical approximation is used.
20 | 	/// @returns The reconstructed mesh.
21 | 	Mesh MarchCube(
22 | 		Fun3s const& sdf,
23 | 		Rect3 const& domain,
24 | 		Vec3 const& cubeSize,
25 | 		double isoLevel = 0,
26 | 		Fun3v sdfGrad = nullptr);
27 | }


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/lib/MeshReconstruction.cpp:
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 1 | #include "MeshReconstruction.h"
 2 | #include "Cube.h"
 3 | #include "Triangulation.h"
 4 | 
 5 | using namespace MeshReconstruction;
 6 | using namespace std;
 7 | 
 8 | // Adapted from here: http://paulbourke.net/geometry/polygonise/
 9 | 
10 | namespace
11 | {
12 | 	Vec3 NumGrad(Fun3s const& f, Vec3 const& p)
13 | 	{
14 | 		auto const Eps = 1e-6;
15 | 		Vec3 epsX{ Eps, 0, 0 }, epsY{ 0, Eps, 0 }, epsZ{ 0, 0, Eps };
16 | 		auto gx = (f(p + epsX) - f(p - epsX)) / 2;
17 | 		auto gy = (f(p + epsY) - f(p - epsY)) / 2;
18 | 		auto gz = (f(p + epsZ) - f(p - epsZ)) / 2;
19 | 		return { gx, gy, gz };
20 | 	}
21 | }
22 | 
23 | Mesh MeshReconstruction::MarchCube(Fun3s const& sdf, Rect3 const& domain)
24 | {
25 | 	auto const NumCubes = 50;
26 | 	auto cubeSize = domain.size * (1.0 / NumCubes);
27 | 
28 | 	return MarchCube(sdf, domain, cubeSize);
29 | }
30 | 
31 | Mesh MeshReconstruction::MarchCube(
32 | 	Fun3s const& sdf,
33 | 	Rect3 const& domain,
34 | 	Vec3 const& cubeSize,
35 | 	double isoLevel,
36 | 	Fun3v sdfGrad)
37 | {
38 | 	// Default value.
39 | 	sdfGrad = sdfGrad == nullptr
40 | 		? [&sdf](Vec3 const& p) { return NumGrad(sdf, p); }
41 | 		: sdfGrad;
42 | 
43 | 	auto const NumX = static_cast<int>(ceil(domain.size.x / cubeSize.x));
44 | 	auto const NumY = static_cast<int>(ceil(domain.size.y / cubeSize.y));
45 | 	auto const NumZ = static_cast<int>(ceil(domain.size.z / cubeSize.z));
46 | 
47 | 	auto const HalfCubeDiag = cubeSize.Norm() / 2.0;
48 | 	auto const HalfCubeSize = cubeSize * 0.5;
49 | 
50 | 	Mesh mesh;
51 | 
52 | 	for (auto ix = 0; ix < NumX; ++ix)
53 | 	{
54 | 		auto x = domain.min.x + ix * cubeSize.x;
55 | 			
56 | 		for (auto iy = 0; iy < NumY; ++iy)
57 | 		{
58 | 			auto y = domain.min.y + iy * cubeSize.y;
59 | 			
60 | 			for (auto iz = 0; iz < NumZ; ++iz)
61 | 			{
62 | 				auto z = domain.min.z + iz * cubeSize.z;
63 | 				Vec3 min{ x, y, z };
64 | 
65 | 				// Process only if cube lies within narrow band around surface.
66 | 				auto cubeCenter = min + HalfCubeSize;
67 | 				auto dist = abs(sdf(cubeCenter) - isoLevel);
68 | 				if (dist > HalfCubeDiag) continue;
69 | 
70 | 				Cube cube({ min, cubeSize }, sdf);
71 | 				auto intersect = cube.Intersect(isoLevel);
72 | 				Triangulate(intersect, sdfGrad, mesh);
73 | 			}
74 | 		}
75 | 	}
76 | 
77 | 	return mesh;
78 | }
79 | 


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/lib/Cube.cpp:
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  1 | #include "Cube.h"
  2 | 
  3 | using namespace MeshReconstruction;
  4 | 
  5 | namespace
  6 | {
  7 | 	// Cube has 8 vertices. Each vertex can have positive or negative sign.
  8 | 	// 2^8 = 256 possible configurations mapped to intersected edges in each case.
  9 | 	// The 12 edges are numbered as 1, 2, 4, ..., 2048 and are stored as a 12-bit bitstring for each configuration.
 10 | 	const int signConfigToIntersectedEdges[256] = {
 11 | 		0x0  , 0x109, 0x203, 0x30a, 0x406, 0x50f, 0x605, 0x70c,
 12 | 		0x80c, 0x905, 0xa0f, 0xb06, 0xc0a, 0xd03, 0xe09, 0xf00,
 13 | 		0x190, 0x99 , 0x393, 0x29a, 0x596, 0x49f, 0x795, 0x69c,
 14 | 		0x99c, 0x895, 0xb9f, 0xa96, 0xd9a, 0xc93, 0xf99, 0xe90,
 15 | 		0x230, 0x339, 0x33 , 0x13a, 0x636, 0x73f, 0x435, 0x53c,
 16 | 		0xa3c, 0xb35, 0x83f, 0x936, 0xe3a, 0xf33, 0xc39, 0xd30,
 17 | 		0x3a0, 0x2a9, 0x1a3, 0xaa , 0x7a6, 0x6af, 0x5a5, 0x4ac,
 18 | 		0xbac, 0xaa5, 0x9af, 0x8a6, 0xfaa, 0xea3, 0xda9, 0xca0,
 19 | 		0x460, 0x569, 0x663, 0x76a, 0x66 , 0x16f, 0x265, 0x36c,
 20 | 		0xc6c, 0xd65, 0xe6f, 0xf66, 0x86a, 0x963, 0xa69, 0xb60,
 21 | 		0x5f0, 0x4f9, 0x7f3, 0x6fa, 0x1f6, 0xff , 0x3f5, 0x2fc,
 22 | 		0xdfc, 0xcf5, 0xfff, 0xef6, 0x9fa, 0x8f3, 0xbf9, 0xaf0,
 23 | 		0x650, 0x759, 0x453, 0x55a, 0x256, 0x35f, 0x55 , 0x15c,
 24 | 		0xe5c, 0xf55, 0xc5f, 0xd56, 0xa5a, 0xb53, 0x859, 0x950,
 25 | 		0x7c0, 0x6c9, 0x5c3, 0x4ca, 0x3c6, 0x2cf, 0x1c5, 0xcc ,
 26 | 		0xfcc, 0xec5, 0xdcf, 0xcc6, 0xbca, 0xac3, 0x9c9, 0x8c0,
 27 | 		0x8c0, 0x9c9, 0xac3, 0xbca, 0xcc6, 0xdcf, 0xec5, 0xfcc,
 28 | 		0xcc , 0x1c5, 0x2cf, 0x3c6, 0x4ca, 0x5c3, 0x6c9, 0x7c0,
 29 | 		0x950, 0x859, 0xb53, 0xa5a, 0xd56, 0xc5f, 0xf55, 0xe5c,
 30 | 		0x15c, 0x55 , 0x35f, 0x256, 0x55a, 0x453, 0x759, 0x650,
 31 | 		0xaf0, 0xbf9, 0x8f3, 0x9fa, 0xef6, 0xfff, 0xcf5, 0xdfc,
 32 | 		0x2fc, 0x3f5, 0xff , 0x1f6, 0x6fa, 0x7f3, 0x4f9, 0x5f0,
 33 | 		0xb60, 0xa69, 0x963, 0x86a, 0xf66, 0xe6f, 0xd65, 0xc6c,
 34 | 		0x36c, 0x265, 0x16f, 0x66 , 0x76a, 0x663, 0x569, 0x460,
 35 | 		0xca0, 0xda9, 0xea3, 0xfaa, 0x8a6, 0x9af, 0xaa5, 0xbac,
 36 | 		0x4ac, 0x5a5, 0x6af, 0x7a6, 0xaa , 0x1a3, 0x2a9, 0x3a0,
 37 | 		0xd30, 0xc39, 0xf33, 0xe3a, 0x936, 0x83f, 0xb35, 0xa3c,
 38 | 		0x53c, 0x435, 0x73f, 0x636, 0x13a, 0x33 , 0x339, 0x230,
 39 | 		0xe90, 0xf99, 0xc93, 0xd9a, 0xa96, 0xb9f, 0x895, 0x99c,
 40 | 		0x69c, 0x795, 0x49f, 0x596, 0x29a, 0x393, 0x99 , 0x190,
 41 | 		0xf00, 0xe09, 0xd03, 0xc0a, 0xb06, 0xa0f, 0x905, 0x80c,
 42 | 		0x70c, 0x605, 0x50f, 0x406, 0x30a, 0x203, 0x109, 0x0 };
 43 | 
 44 | 	struct Edge
 45 | 	{
 46 | 		int edgeFlag : 12; // flag: 1, 2, 4, ... 2048
 47 | 		int vert0; // 0-7
 48 | 		int vert1; // 0-7
 49 | 	};
 50 | 
 51 | 	const Edge edges[12] =
 52 | 	{
 53 | 		{ 1, 0, 1 }, // edge 0
 54 | 		{ 2, 1, 2 }, // edge 1
 55 | 		{ 4, 2, 3 }, // ...
 56 | 		{ 8, 3, 0 },
 57 | 		{ 16, 4, 5 },
 58 | 		{ 32, 5, 6 },
 59 | 		{ 64, 6, 7 },
 60 | 		{ 128, 7, 4 },
 61 | 		{ 256, 0, 4 },
 62 | 		{ 512, 1, 5 },
 63 | 		{ 1024, 2, 6 },
 64 | 		{ 2048, 3, 7 } // edge 11
 65 | 	};
 66 | }
 67 | 
 68 | Vec3 Cube::LerpVertex(double isoLevel, int i1, int i2) const
 69 | {
 70 | 	auto const Eps = 1e-5;
 71 | 	auto const v1 = sdf[i1];
 72 | 	auto const v2 = sdf[i2];
 73 | 	auto const& p1 = pos[i1];
 74 | 	auto const& p2 = pos[i2];
 75 | 
 76 | 	if (abs(isoLevel - v1) < Eps) return p1;
 77 | 	if (abs(isoLevel - v2) < Eps) return p2;
 78 | 	if (abs(v1 - v2) < Eps) return p1;
 79 | 
 80 | 	auto mu = (isoLevel - v1) / (v2 - v1);
 81 | 	return p1 + (p2 - p1)*mu;
 82 | }
 83 | 
 84 | Cube::Cube(Rect3 const& space, Fun3s const& sdf)
 85 | {
 86 | 	auto mx = space.min.x;
 87 | 	auto my = space.min.y;
 88 | 	auto mz = space.min.z;
 89 | 
 90 | 	auto sx = space.size.x;
 91 | 	auto sy = space.size.y;
 92 | 	auto sz = space.size.z;
 93 | 
 94 | 	pos[0] = space.min;
 95 | 	pos[1] = { mx + sx, my, mz };
 96 | 	pos[2] = { mx + sx, my, mz + sz };
 97 | 	pos[3] = { mx, my, mz + sz };
 98 | 	pos[4] = { mx, my + sy, mz };
 99 | 	pos[5] = { mx + sx, my + sy, mz };
100 | 	pos[6] = { mx + sx, my + sy, mz + sz };
101 | 	pos[7] = { mx, my + sy, mz + sz };
102 | 
103 | 	for (auto i = 0; i < 8; ++i)
104 | 	{
105 | 		auto sd = sdf(pos[i]);
106 | 		if (sd == 0) sd += 1e-6;
107 | 		this->sdf[i] = sd;
108 | 	}
109 | }
110 | 
111 | int Cube::SignConfig(double isoLevel) const
112 | {
113 | 	auto edgeIndex = 0;
114 | 
115 | 	for (auto i = 0; i < 8; ++i)
116 | 	{
117 | 		if (sdf[i] < isoLevel)
118 | 		{
119 | 			edgeIndex |= 1 << i;
120 | 		}
121 | 	}
122 | 
123 | 	return edgeIndex;
124 | }
125 | 
126 | IntersectInfo Cube::Intersect(double iso) const
127 | {
128 | 	// idea:
129 | 	// from signs at 8 corners of cube a sign configuration (256 possible ones) is computed
130 | 	// this configuration can be used to index into a table that tells which of the 12 edges are intersected
131 | 	// find vertices adjacent to edges and interpolate cut vertex and store it in IntersectionInfo object
132 | 
133 | 	IntersectInfo intersect;
134 | 	intersect.signConfig = SignConfig(iso);
135 | 
136 | 	for (auto e = 0; e<12; ++e)
137 | 	{
138 | 		if (signConfigToIntersectedEdges[intersect.signConfig] & edges[e].edgeFlag)
139 | 		{
140 | 			auto v0 = edges[e].vert0;
141 | 			auto v1 = edges[e].vert1;
142 | 			auto vert = LerpVertex(iso, v0, v1);
143 | 			intersect.edgeVertIndices[e] = vert;
144 | 		}
145 | 	}
146 | 
147 | 	return intersect;
148 | }
149 | 


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/lib/Triangulation.cpp:
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  1 | #include "Triangulation.h"
  2 | 
  3 | namespace
  4 | {
  5 | 	// Indices into vertex buffer (0 - 11).
  6 | 	// Three successive entries make up one triangle.
  7 | 	// -1 means unused.
  8 | 	const int signConfigToTriangles[256][16] =
  9 | 		{ { -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 10 | 		{ 0, 8, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 11 | 		{ 0, 1, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 12 | 		{ 1, 8, 3, 9, 8, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 13 | 		{ 1, 2, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 14 | 		{ 0, 8, 3, 1, 2, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 15 | 		{ 9, 2, 10, 0, 2, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 16 | 		{ 2, 8, 3, 2, 10, 8, 10, 9, 8, -1, -1, -1, -1, -1, -1, -1 },
 17 | 		{ 3, 11, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 18 | 		{ 0, 11, 2, 8, 11, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 19 | 		{ 1, 9, 0, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 20 | 		{ 1, 11, 2, 1, 9, 11, 9, 8, 11, -1, -1, -1, -1, -1, -1, -1 },
 21 | 		{ 3, 10, 1, 11, 10, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 22 | 		{ 0, 10, 1, 0, 8, 10, 8, 11, 10, -1, -1, -1, -1, -1, -1, -1 },
 23 | 		{ 3, 9, 0, 3, 11, 9, 11, 10, 9, -1, -1, -1, -1, -1, -1, -1 },
 24 | 		{ 9, 8, 10, 10, 8, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 25 | 		{ 4, 7, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 26 | 		{ 4, 3, 0, 7, 3, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 27 | 		{ 0, 1, 9, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 28 | 		{ 4, 1, 9, 4, 7, 1, 7, 3, 1, -1, -1, -1, -1, -1, -1, -1 },
 29 | 		{ 1, 2, 10, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 30 | 		{ 3, 4, 7, 3, 0, 4, 1, 2, 10, -1, -1, -1, -1, -1, -1, -1 },
 31 | 		{ 9, 2, 10, 9, 0, 2, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1 },
 32 | 		{ 2, 10, 9, 2, 9, 7, 2, 7, 3, 7, 9, 4, -1, -1, -1, -1 },
 33 | 		{ 8, 4, 7, 3, 11, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 34 | 		{ 11, 4, 7, 11, 2, 4, 2, 0, 4, -1, -1, -1, -1, -1, -1, -1 },
 35 | 		{ 9, 0, 1, 8, 4, 7, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1 },
 36 | 		{ 4, 7, 11, 9, 4, 11, 9, 11, 2, 9, 2, 1, -1, -1, -1, -1 },
 37 | 		{ 3, 10, 1, 3, 11, 10, 7, 8, 4, -1, -1, -1, -1, -1, -1, -1 },
 38 | 		{ 1, 11, 10, 1, 4, 11, 1, 0, 4, 7, 11, 4, -1, -1, -1, -1 },
 39 | 		{ 4, 7, 8, 9, 0, 11, 9, 11, 10, 11, 0, 3, -1, -1, -1, -1 },
 40 | 		{ 4, 7, 11, 4, 11, 9, 9, 11, 10, -1, -1, -1, -1, -1, -1, -1 },
 41 | 		{ 9, 5, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 42 | 		{ 9, 5, 4, 0, 8, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 43 | 		{ 0, 5, 4, 1, 5, 0, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 44 | 		{ 8, 5, 4, 8, 3, 5, 3, 1, 5, -1, -1, -1, -1, -1, -1, -1 },
 45 | 		{ 1, 2, 10, 9, 5, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 46 | 		{ 3, 0, 8, 1, 2, 10, 4, 9, 5, -1, -1, -1, -1, -1, -1, -1 },
 47 | 		{ 5, 2, 10, 5, 4, 2, 4, 0, 2, -1, -1, -1, -1, -1, -1, -1 },
 48 | 		{ 2, 10, 5, 3, 2, 5, 3, 5, 4, 3, 4, 8, -1, -1, -1, -1 },
 49 | 		{ 9, 5, 4, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 50 | 		{ 0, 11, 2, 0, 8, 11, 4, 9, 5, -1, -1, -1, -1, -1, -1, -1 },
 51 | 		{ 0, 5, 4, 0, 1, 5, 2, 3, 11, -1, -1, -1, -1, -1, -1, -1 },
 52 | 		{ 2, 1, 5, 2, 5, 8, 2, 8, 11, 4, 8, 5, -1, -1, -1, -1 },
 53 | 		{ 10, 3, 11, 10, 1, 3, 9, 5, 4, -1, -1, -1, -1, -1, -1, -1 },
 54 | 		{ 4, 9, 5, 0, 8, 1, 8, 10, 1, 8, 11, 10, -1, -1, -1, -1 },
 55 | 		{ 5, 4, 0, 5, 0, 11, 5, 11, 10, 11, 0, 3, -1, -1, -1, -1 },
 56 | 		{ 5, 4, 8, 5, 8, 10, 10, 8, 11, -1, -1, -1, -1, -1, -1, -1 },
 57 | 		{ 9, 7, 8, 5, 7, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 58 | 		{ 9, 3, 0, 9, 5, 3, 5, 7, 3, -1, -1, -1, -1, -1, -1, -1 },
 59 | 		{ 0, 7, 8, 0, 1, 7, 1, 5, 7, -1, -1, -1, -1, -1, -1, -1 },
 60 | 		{ 1, 5, 3, 3, 5, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 61 | 		{ 9, 7, 8, 9, 5, 7, 10, 1, 2, -1, -1, -1, -1, -1, -1, -1 },
 62 | 		{ 10, 1, 2, 9, 5, 0, 5, 3, 0, 5, 7, 3, -1, -1, -1, -1 },
 63 | 		{ 8, 0, 2, 8, 2, 5, 8, 5, 7, 10, 5, 2, -1, -1, -1, -1 },
 64 | 		{ 2, 10, 5, 2, 5, 3, 3, 5, 7, -1, -1, -1, -1, -1, -1, -1 },
 65 | 		{ 7, 9, 5, 7, 8, 9, 3, 11, 2, -1, -1, -1, -1, -1, -1, -1 },
 66 | 		{ 9, 5, 7, 9, 7, 2, 9, 2, 0, 2, 7, 11, -1, -1, -1, -1 },
 67 | 		{ 2, 3, 11, 0, 1, 8, 1, 7, 8, 1, 5, 7, -1, -1, -1, -1 },
 68 | 		{ 11, 2, 1, 11, 1, 7, 7, 1, 5, -1, -1, -1, -1, -1, -1, -1 },
 69 | 		{ 9, 5, 8, 8, 5, 7, 10, 1, 3, 10, 3, 11, -1, -1, -1, -1 },
 70 | 		{ 5, 7, 0, 5, 0, 9, 7, 11, 0, 1, 0, 10, 11, 10, 0, -1 },
 71 | 		{ 11, 10, 0, 11, 0, 3, 10, 5, 0, 8, 0, 7, 5, 7, 0, -1 },
 72 | 		{ 11, 10, 5, 7, 11, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 73 | 		{ 10, 6, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 74 | 		{ 0, 8, 3, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 75 | 		{ 9, 0, 1, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 76 | 		{ 1, 8, 3, 1, 9, 8, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1 },
 77 | 		{ 1, 6, 5, 2, 6, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 78 | 		{ 1, 6, 5, 1, 2, 6, 3, 0, 8, -1, -1, -1, -1, -1, -1, -1 },
 79 | 		{ 9, 6, 5, 9, 0, 6, 0, 2, 6, -1, -1, -1, -1, -1, -1, -1 },
 80 | 		{ 5, 9, 8, 5, 8, 2, 5, 2, 6, 3, 2, 8, -1, -1, -1, -1 },
 81 | 		{ 2, 3, 11, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 82 | 		{ 11, 0, 8, 11, 2, 0, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1 },
 83 | 		{ 0, 1, 9, 2, 3, 11, 5, 10, 6, -1, -1, -1, -1, -1, -1, -1 },
 84 | 		{ 5, 10, 6, 1, 9, 2, 9, 11, 2, 9, 8, 11, -1, -1, -1, -1 },
 85 | 		{ 6, 3, 11, 6, 5, 3, 5, 1, 3, -1, -1, -1, -1, -1, -1, -1 },
 86 | 		{ 0, 8, 11, 0, 11, 5, 0, 5, 1, 5, 11, 6, -1, -1, -1, -1 },
 87 | 		{ 3, 11, 6, 0, 3, 6, 0, 6, 5, 0, 5, 9, -1, -1, -1, -1 },
 88 | 		{ 6, 5, 9, 6, 9, 11, 11, 9, 8, -1, -1, -1, -1, -1, -1, -1 },
 89 | 		{ 5, 10, 6, 4, 7, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
 90 | 		{ 4, 3, 0, 4, 7, 3, 6, 5, 10, -1, -1, -1, -1, -1, -1, -1 },
 91 | 		{ 1, 9, 0, 5, 10, 6, 8, 4, 7, -1, -1, -1, -1, -1, -1, -1 },
 92 | 		{ 10, 6, 5, 1, 9, 7, 1, 7, 3, 7, 9, 4, -1, -1, -1, -1 },
 93 | 		{ 6, 1, 2, 6, 5, 1, 4, 7, 8, -1, -1, -1, -1, -1, -1, -1 },
 94 | 		{ 1, 2, 5, 5, 2, 6, 3, 0, 4, 3, 4, 7, -1, -1, -1, -1 },
 95 | 		{ 8, 4, 7, 9, 0, 5, 0, 6, 5, 0, 2, 6, -1, -1, -1, -1 },
 96 | 		{ 7, 3, 9, 7, 9, 4, 3, 2, 9, 5, 9, 6, 2, 6, 9, -1 },
 97 | 		{ 3, 11, 2, 7, 8, 4, 10, 6, 5, -1, -1, -1, -1, -1, -1, -1 },
 98 | 		{ 5, 10, 6, 4, 7, 2, 4, 2, 0, 2, 7, 11, -1, -1, -1, -1 },
 99 | 		{ 0, 1, 9, 4, 7, 8, 2, 3, 11, 5, 10, 6, -1, -1, -1, -1 },
100 | 		{ 9, 2, 1, 9, 11, 2, 9, 4, 11, 7, 11, 4, 5, 10, 6, -1 },
101 | 		{ 8, 4, 7, 3, 11, 5, 3, 5, 1, 5, 11, 6, -1, -1, -1, -1 },
102 | 		{ 5, 1, 11, 5, 11, 6, 1, 0, 11, 7, 11, 4, 0, 4, 11, -1 },
103 | 		{ 0, 5, 9, 0, 6, 5, 0, 3, 6, 11, 6, 3, 8, 4, 7, -1 },
104 | 		{ 6, 5, 9, 6, 9, 11, 4, 7, 9, 7, 11, 9, -1, -1, -1, -1 },
105 | 		{ 10, 4, 9, 6, 4, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
106 | 		{ 4, 10, 6, 4, 9, 10, 0, 8, 3, -1, -1, -1, -1, -1, -1, -1 },
107 | 		{ 10, 0, 1, 10, 6, 0, 6, 4, 0, -1, -1, -1, -1, -1, -1, -1 },
108 | 		{ 8, 3, 1, 8, 1, 6, 8, 6, 4, 6, 1, 10, -1, -1, -1, -1 },
109 | 		{ 1, 4, 9, 1, 2, 4, 2, 6, 4, -1, -1, -1, -1, -1, -1, -1 },
110 | 		{ 3, 0, 8, 1, 2, 9, 2, 4, 9, 2, 6, 4, -1, -1, -1, -1 },
111 | 		{ 0, 2, 4, 4, 2, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
112 | 		{ 8, 3, 2, 8, 2, 4, 4, 2, 6, -1, -1, -1, -1, -1, -1, -1 },
113 | 		{ 10, 4, 9, 10, 6, 4, 11, 2, 3, -1, -1, -1, -1, -1, -1, -1 },
114 | 		{ 0, 8, 2, 2, 8, 11, 4, 9, 10, 4, 10, 6, -1, -1, -1, -1 },
115 | 		{ 3, 11, 2, 0, 1, 6, 0, 6, 4, 6, 1, 10, -1, -1, -1, -1 },
116 | 		{ 6, 4, 1, 6, 1, 10, 4, 8, 1, 2, 1, 11, 8, 11, 1, -1 },
117 | 		{ 9, 6, 4, 9, 3, 6, 9, 1, 3, 11, 6, 3, -1, -1, -1, -1 },
118 | 		{ 8, 11, 1, 8, 1, 0, 11, 6, 1, 9, 1, 4, 6, 4, 1, -1 },
119 | 		{ 3, 11, 6, 3, 6, 0, 0, 6, 4, -1, -1, -1, -1, -1, -1, -1 },
120 | 		{ 6, 4, 8, 11, 6, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
121 | 		{ 7, 10, 6, 7, 8, 10, 8, 9, 10, -1, -1, -1, -1, -1, -1, -1 },
122 | 		{ 0, 7, 3, 0, 10, 7, 0, 9, 10, 6, 7, 10, -1, -1, -1, -1 },
123 | 		{ 10, 6, 7, 1, 10, 7, 1, 7, 8, 1, 8, 0, -1, -1, -1, -1 },
124 | 		{ 10, 6, 7, 10, 7, 1, 1, 7, 3, -1, -1, -1, -1, -1, -1, -1 },
125 | 		{ 1, 2, 6, 1, 6, 8, 1, 8, 9, 8, 6, 7, -1, -1, -1, -1 },
126 | 		{ 2, 6, 9, 2, 9, 1, 6, 7, 9, 0, 9, 3, 7, 3, 9, -1 },
127 | 		{ 7, 8, 0, 7, 0, 6, 6, 0, 2, -1, -1, -1, -1, -1, -1, -1 },
128 | 		{ 7, 3, 2, 6, 7, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
129 | 		{ 2, 3, 11, 10, 6, 8, 10, 8, 9, 8, 6, 7, -1, -1, -1, -1 },
130 | 		{ 2, 0, 7, 2, 7, 11, 0, 9, 7, 6, 7, 10, 9, 10, 7, -1 },
131 | 		{ 1, 8, 0, 1, 7, 8, 1, 10, 7, 6, 7, 10, 2, 3, 11, -1 },
132 | 		{ 11, 2, 1, 11, 1, 7, 10, 6, 1, 6, 7, 1, -1, -1, -1, -1 },
133 | 		{ 8, 9, 6, 8, 6, 7, 9, 1, 6, 11, 6, 3, 1, 3, 6, -1 },
134 | 		{ 0, 9, 1, 11, 6, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
135 | 		{ 7, 8, 0, 7, 0, 6, 3, 11, 0, 11, 6, 0, -1, -1, -1, -1 },
136 | 		{ 7, 11, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
137 | 		{ 7, 6, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
138 | 		{ 3, 0, 8, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
139 | 		{ 0, 1, 9, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
140 | 		{ 8, 1, 9, 8, 3, 1, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1 },
141 | 		{ 10, 1, 2, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
142 | 		{ 1, 2, 10, 3, 0, 8, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1 },
143 | 		{ 2, 9, 0, 2, 10, 9, 6, 11, 7, -1, -1, -1, -1, -1, -1, -1 },
144 | 		{ 6, 11, 7, 2, 10, 3, 10, 8, 3, 10, 9, 8, -1, -1, -1, -1 },
145 | 		{ 7, 2, 3, 6, 2, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
146 | 		{ 7, 0, 8, 7, 6, 0, 6, 2, 0, -1, -1, -1, -1, -1, -1, -1 },
147 | 		{ 2, 7, 6, 2, 3, 7, 0, 1, 9, -1, -1, -1, -1, -1, -1, -1 },
148 | 		{ 1, 6, 2, 1, 8, 6, 1, 9, 8, 8, 7, 6, -1, -1, -1, -1 },
149 | 		{ 10, 7, 6, 10, 1, 7, 1, 3, 7, -1, -1, -1, -1, -1, -1, -1 },
150 | 		{ 10, 7, 6, 1, 7, 10, 1, 8, 7, 1, 0, 8, -1, -1, -1, -1 },
151 | 		{ 0, 3, 7, 0, 7, 10, 0, 10, 9, 6, 10, 7, -1, -1, -1, -1 },
152 | 		{ 7, 6, 10, 7, 10, 8, 8, 10, 9, -1, -1, -1, -1, -1, -1, -1 },
153 | 		{ 6, 8, 4, 11, 8, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
154 | 		{ 3, 6, 11, 3, 0, 6, 0, 4, 6, -1, -1, -1, -1, -1, -1, -1 },
155 | 		{ 8, 6, 11, 8, 4, 6, 9, 0, 1, -1, -1, -1, -1, -1, -1, -1 },
156 | 		{ 9, 4, 6, 9, 6, 3, 9, 3, 1, 11, 3, 6, -1, -1, -1, -1 },
157 | 		{ 6, 8, 4, 6, 11, 8, 2, 10, 1, -1, -1, -1, -1, -1, -1, -1 },
158 | 		{ 1, 2, 10, 3, 0, 11, 0, 6, 11, 0, 4, 6, -1, -1, -1, -1 },
159 | 		{ 4, 11, 8, 4, 6, 11, 0, 2, 9, 2, 10, 9, -1, -1, -1, -1 },
160 | 		{ 10, 9, 3, 10, 3, 2, 9, 4, 3, 11, 3, 6, 4, 6, 3, -1 },
161 | 		{ 8, 2, 3, 8, 4, 2, 4, 6, 2, -1, -1, -1, -1, -1, -1, -1 },
162 | 		{ 0, 4, 2, 4, 6, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
163 | 		{ 1, 9, 0, 2, 3, 4, 2, 4, 6, 4, 3, 8, -1, -1, -1, -1 },
164 | 		{ 1, 9, 4, 1, 4, 2, 2, 4, 6, -1, -1, -1, -1, -1, -1, -1 },
165 | 		{ 8, 1, 3, 8, 6, 1, 8, 4, 6, 6, 10, 1, -1, -1, -1, -1 },
166 | 		{ 10, 1, 0, 10, 0, 6, 6, 0, 4, -1, -1, -1, -1, -1, -1, -1 },
167 | 		{ 4, 6, 3, 4, 3, 8, 6, 10, 3, 0, 3, 9, 10, 9, 3, -1 },
168 | 		{ 10, 9, 4, 6, 10, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
169 | 		{ 4, 9, 5, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
170 | 		{ 0, 8, 3, 4, 9, 5, 11, 7, 6, -1, -1, -1, -1, -1, -1, -1 },
171 | 		{ 5, 0, 1, 5, 4, 0, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1 },
172 | 		{ 11, 7, 6, 8, 3, 4, 3, 5, 4, 3, 1, 5, -1, -1, -1, -1 },
173 | 		{ 9, 5, 4, 10, 1, 2, 7, 6, 11, -1, -1, -1, -1, -1, -1, -1 },
174 | 		{ 6, 11, 7, 1, 2, 10, 0, 8, 3, 4, 9, 5, -1, -1, -1, -1 },
175 | 		{ 7, 6, 11, 5, 4, 10, 4, 2, 10, 4, 0, 2, -1, -1, -1, -1 },
176 | 		{ 3, 4, 8, 3, 5, 4, 3, 2, 5, 10, 5, 2, 11, 7, 6, -1 },
177 | 		{ 7, 2, 3, 7, 6, 2, 5, 4, 9, -1, -1, -1, -1, -1, -1, -1 },
178 | 		{ 9, 5, 4, 0, 8, 6, 0, 6, 2, 6, 8, 7, -1, -1, -1, -1 },
179 | 		{ 3, 6, 2, 3, 7, 6, 1, 5, 0, 5, 4, 0, -1, -1, -1, -1 },
180 | 		{ 6, 2, 8, 6, 8, 7, 2, 1, 8, 4, 8, 5, 1, 5, 8, -1 },
181 | 		{ 9, 5, 4, 10, 1, 6, 1, 7, 6, 1, 3, 7, -1, -1, -1, -1 },
182 | 		{ 1, 6, 10, 1, 7, 6, 1, 0, 7, 8, 7, 0, 9, 5, 4, -1 },
183 | 		{ 4, 0, 10, 4, 10, 5, 0, 3, 10, 6, 10, 7, 3, 7, 10, -1 },
184 | 		{ 7, 6, 10, 7, 10, 8, 5, 4, 10, 4, 8, 10, -1, -1, -1, -1 },
185 | 		{ 6, 9, 5, 6, 11, 9, 11, 8, 9, -1, -1, -1, -1, -1, -1, -1 },
186 | 		{ 3, 6, 11, 0, 6, 3, 0, 5, 6, 0, 9, 5, -1, -1, -1, -1 },
187 | 		{ 0, 11, 8, 0, 5, 11, 0, 1, 5, 5, 6, 11, -1, -1, -1, -1 },
188 | 		{ 6, 11, 3, 6, 3, 5, 5, 3, 1, -1, -1, -1, -1, -1, -1, -1 },
189 | 		{ 1, 2, 10, 9, 5, 11, 9, 11, 8, 11, 5, 6, -1, -1, -1, -1 },
190 | 		{ 0, 11, 3, 0, 6, 11, 0, 9, 6, 5, 6, 9, 1, 2, 10, -1 },
191 | 		{ 11, 8, 5, 11, 5, 6, 8, 0, 5, 10, 5, 2, 0, 2, 5, -1 },
192 | 		{ 6, 11, 3, 6, 3, 5, 2, 10, 3, 10, 5, 3, -1, -1, -1, -1 },
193 | 		{ 5, 8, 9, 5, 2, 8, 5, 6, 2, 3, 8, 2, -1, -1, -1, -1 },
194 | 		{ 9, 5, 6, 9, 6, 0, 0, 6, 2, -1, -1, -1, -1, -1, -1, -1 },
195 | 		{ 1, 5, 8, 1, 8, 0, 5, 6, 8, 3, 8, 2, 6, 2, 8, -1 },
196 | 		{ 1, 5, 6, 2, 1, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
197 | 		{ 1, 3, 6, 1, 6, 10, 3, 8, 6, 5, 6, 9, 8, 9, 6, -1 },
198 | 		{ 10, 1, 0, 10, 0, 6, 9, 5, 0, 5, 6, 0, -1, -1, -1, -1 },
199 | 		{ 0, 3, 8, 5, 6, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
200 | 		{ 10, 5, 6, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
201 | 		{ 11, 5, 10, 7, 5, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
202 | 		{ 11, 5, 10, 11, 7, 5, 8, 3, 0, -1, -1, -1, -1, -1, -1, -1 },
203 | 		{ 5, 11, 7, 5, 10, 11, 1, 9, 0, -1, -1, -1, -1, -1, -1, -1 },
204 | 		{ 10, 7, 5, 10, 11, 7, 9, 8, 1, 8, 3, 1, -1, -1, -1, -1 },
205 | 		{ 11, 1, 2, 11, 7, 1, 7, 5, 1, -1, -1, -1, -1, -1, -1, -1 },
206 | 		{ 0, 8, 3, 1, 2, 7, 1, 7, 5, 7, 2, 11, -1, -1, -1, -1 },
207 | 		{ 9, 7, 5, 9, 2, 7, 9, 0, 2, 2, 11, 7, -1, -1, -1, -1 },
208 | 		{ 7, 5, 2, 7, 2, 11, 5, 9, 2, 3, 2, 8, 9, 8, 2, -1 },
209 | 		{ 2, 5, 10, 2, 3, 5, 3, 7, 5, -1, -1, -1, -1, -1, -1, -1 },
210 | 		{ 8, 2, 0, 8, 5, 2, 8, 7, 5, 10, 2, 5, -1, -1, -1, -1 },
211 | 		{ 9, 0, 1, 5, 10, 3, 5, 3, 7, 3, 10, 2, -1, -1, -1, -1 },
212 | 		{ 9, 8, 2, 9, 2, 1, 8, 7, 2, 10, 2, 5, 7, 5, 2, -1 },
213 | 		{ 1, 3, 5, 3, 7, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
214 | 		{ 0, 8, 7, 0, 7, 1, 1, 7, 5, -1, -1, -1, -1, -1, -1, -1 },
215 | 		{ 9, 0, 3, 9, 3, 5, 5, 3, 7, -1, -1, -1, -1, -1, -1, -1 },
216 | 		{ 9, 8, 7, 5, 9, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
217 | 		{ 5, 8, 4, 5, 10, 8, 10, 11, 8, -1, -1, -1, -1, -1, -1, -1 },
218 | 		{ 5, 0, 4, 5, 11, 0, 5, 10, 11, 11, 3, 0, -1, -1, -1, -1 },
219 | 		{ 0, 1, 9, 8, 4, 10, 8, 10, 11, 10, 4, 5, -1, -1, -1, -1 },
220 | 		{ 10, 11, 4, 10, 4, 5, 11, 3, 4, 9, 4, 1, 3, 1, 4, -1 },
221 | 		{ 2, 5, 1, 2, 8, 5, 2, 11, 8, 4, 5, 8, -1, -1, -1, -1 },
222 | 		{ 0, 4, 11, 0, 11, 3, 4, 5, 11, 2, 11, 1, 5, 1, 11, -1 },
223 | 		{ 0, 2, 5, 0, 5, 9, 2, 11, 5, 4, 5, 8, 11, 8, 5, -1 },
224 | 		{ 9, 4, 5, 2, 11, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
225 | 		{ 2, 5, 10, 3, 5, 2, 3, 4, 5, 3, 8, 4, -1, -1, -1, -1 },
226 | 		{ 5, 10, 2, 5, 2, 4, 4, 2, 0, -1, -1, -1, -1, -1, -1, -1 },
227 | 		{ 3, 10, 2, 3, 5, 10, 3, 8, 5, 4, 5, 8, 0, 1, 9, -1 },
228 | 		{ 5, 10, 2, 5, 2, 4, 1, 9, 2, 9, 4, 2, -1, -1, -1, -1 },
229 | 		{ 8, 4, 5, 8, 5, 3, 3, 5, 1, -1, -1, -1, -1, -1, -1, -1 },
230 | 		{ 0, 4, 5, 1, 0, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
231 | 		{ 8, 4, 5, 8, 5, 3, 9, 0, 5, 0, 3, 5, -1, -1, -1, -1 },
232 | 		{ 9, 4, 5, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
233 | 		{ 4, 11, 7, 4, 9, 11, 9, 10, 11, -1, -1, -1, -1, -1, -1, -1 },
234 | 		{ 0, 8, 3, 4, 9, 7, 9, 11, 7, 9, 10, 11, -1, -1, -1, -1 },
235 | 		{ 1, 10, 11, 1, 11, 4, 1, 4, 0, 7, 4, 11, -1, -1, -1, -1 },
236 | 		{ 3, 1, 4, 3, 4, 8, 1, 10, 4, 7, 4, 11, 10, 11, 4, -1 },
237 | 		{ 4, 11, 7, 9, 11, 4, 9, 2, 11, 9, 1, 2, -1, -1, -1, -1 },
238 | 		{ 9, 7, 4, 9, 11, 7, 9, 1, 11, 2, 11, 1, 0, 8, 3, -1 },
239 | 		{ 11, 7, 4, 11, 4, 2, 2, 4, 0, -1, -1, -1, -1, -1, -1, -1 },
240 | 		{ 11, 7, 4, 11, 4, 2, 8, 3, 4, 3, 2, 4, -1, -1, -1, -1 },
241 | 		{ 2, 9, 10, 2, 7, 9, 2, 3, 7, 7, 4, 9, -1, -1, -1, -1 },
242 | 		{ 9, 10, 7, 9, 7, 4, 10, 2, 7, 8, 7, 0, 2, 0, 7, -1 },
243 | 		{ 3, 7, 10, 3, 10, 2, 7, 4, 10, 1, 10, 0, 4, 0, 10, -1 },
244 | 		{ 1, 10, 2, 8, 7, 4, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
245 | 		{ 4, 9, 1, 4, 1, 7, 7, 1, 3, -1, -1, -1, -1, -1, -1, -1 },
246 | 		{ 4, 9, 1, 4, 1, 7, 0, 8, 1, 8, 7, 1, -1, -1, -1, -1 },
247 | 		{ 4, 0, 3, 7, 4, 3, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
248 | 		{ 4, 8, 7, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
249 | 		{ 9, 10, 8, 10, 11, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
250 | 		{ 3, 0, 9, 3, 9, 11, 11, 9, 10, -1, -1, -1, -1, -1, -1, -1 },
251 | 		{ 0, 1, 10, 0, 10, 8, 8, 10, 11, -1, -1, -1, -1, -1, -1, -1 },
252 | 		{ 3, 1, 10, 11, 3, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
253 | 		{ 1, 2, 11, 1, 11, 9, 9, 11, 8, -1, -1, -1, -1, -1, -1, -1 },
254 | 		{ 3, 0, 9, 3, 9, 11, 1, 2, 9, 2, 11, 9, -1, -1, -1, -1 },
255 | 		{ 0, 2, 11, 8, 0, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
256 | 		{ 3, 2, 11, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
257 | 		{ 2, 3, 8, 2, 8, 10, 10, 8, 9, -1, -1, -1, -1, -1, -1, -1 },
258 | 		{ 9, 10, 2, 0, 9, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
259 | 		{ 2, 3, 8, 2, 8, 10, 0, 1, 8, 1, 10, 8, -1, -1, -1, -1 },
260 | 		{ 1, 10, 2, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
261 | 		{ 1, 3, 8, 9, 1, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
262 | 		{ 0, 9, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
263 | 		{ 0, 3, 8, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 },
264 | 		{ -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1 } };
265 | }
266 | 
267 | /// Given a grid cube and an isolevel the triangles (5 max)
268 | /// required to represent the isosurface in the cube are computed.
269 | void MeshReconstruction::Triangulate(
270 | 	IntersectInfo const& intersect,
271 | 	Fun3v const& grad,
272 | 	Mesh& mesh)
273 | {
274 | 	// Cube is entirely in/out of the surface. Generate no triangles.
275 | 	if (intersect.signConfig == 0 || intersect.signConfig == 255) return;
276 | 
277 | 	auto const& tri = signConfigToTriangles[intersect.signConfig];
278 | 
279 | 	for (auto i = 0; tri[i] != -1; i += 3)
280 | 	{
281 | 		auto const& v0 = intersect.edgeVertIndices[tri[i]];
282 | 		auto const& v1 = intersect.edgeVertIndices[tri[i + 1]];
283 | 		auto const& v2 = intersect.edgeVertIndices[tri[i + 2]];
284 | 
285 | 		mesh.vertices.push_back(v0);
286 | 		mesh.vertices.push_back(v1);
287 | 		mesh.vertices.push_back(v2);
288 | 
289 | 		auto normal0 = grad(v0).Normalized();
290 | 		auto normal1 = grad(v1).Normalized();
291 | 		auto normal2 = grad(v2).Normalized();
292 | 
293 | 		mesh.vertexNormals.push_back(normal0);
294 | 		mesh.vertexNormals.push_back(normal1);
295 | 		mesh.vertexNormals.push_back(normal2);
296 | 
297 | 		auto last = static_cast<int>(mesh.vertices.size() - 1);
298 | 
299 | 		mesh.triangles.push_back({ last - 2, last - 1, last });
300 | 	}
301 | }


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